Group #20283366 — subgroups

Subgroups — #20283366

Order 6784.

384 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810684	C₂	1	4	yes	maximal, complemented
2	#16810680	C₂	1	4	yes	maximal, complemented
3	#16810650	C₂	1	2	yes	maximal, complemented
4	#16810590	C₂	1	2	yes	maximal
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂²×#88322	C₂²	1	2	yes	—
10	C₄⋊#88322	C₂²	1	8	yes	—
11	C₂²⋊#88322	C₂²	1	8	yes	—
12	#13133128	C₂²	1	2	yes	—
13	#13133128	C₂²	1	2	yes	—
14	#13133128	C₂²	1	2	yes	—
15	C₄×#88322	C₂²	1	2	yes	—
16	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₅₃	C₄	1	2	yes	—
18	C₄×D₄×C₅₃	C₄	1	2	yes	—
19	C₄×D₄×C₅₃	C₄	1	4	yes	—
20	#117×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
24	#41×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₄²×C₅₃	C₂×C₄	1	4	yes	—
37	C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄²×C₅₃	C₂³	1	2	yes	—
41	#11719917	—	2	4	no	—
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	C₂³	1	4	yes	—
44	C₂×#88322	C₂³	1	4	yes	—
45	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₅₃	—	2	2	no	—
48	D₄×C₅₃	—	2	4	no	—
49	D₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	4	yes	—
59	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	#88322	—	2	4	no	—
61	#88322	—	2	4	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	—	2	4	no	—
65	C₂²×C₅₃	#117	1	2	yes	—
66	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	4	no	—
71	C₄×C₅₃	#117	1	2	yes	—
72	C₄×C₅₃	C₂×#41	1	2	yes	—
73	#2923	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	—	4	2	no	—
76	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
77	C₂×C₅₃	#568	1	2	yes	—
78	C₄×#41	D₅₃	1	1	yes	characteristic
79	#433	—	53	4	no	—
80	#432	—	53	4	no	—
81	#424	—	53	2	no	—
82	#408	—	53	2	no	—
83	C₄×M₁₆	—	53	1	no	—
84	C₂×#97	—	53	1	no	—
85	C₅₃	#2923	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
87	C₂×M₁₆	—	53	2	no	—
88	C₂²×C₈	—	53	2	no	—
89	C₄×Q₈	Dic₅₃	1	2	yes	—
90	C₄×D₄	Dic₅₃	1	4	yes	—
91	C₄×D₄	Dic₅₃	1	2	yes	—
92	#117	C₂×D₅₃	1	1	yes	characteristic
93	#117	C₂×D₅₃	1	2	yes	—
94	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₅₃	1	2	yes	—
96	C₄⋊C₈	—	53	8	no	—
97	C₂²⋊C₈	—	53	8	no	—
98	#97	—	53	2	no	—
99	#97	—	53	2	no	—
100	#97	—	53	2	no	—
101	C₄×C₈	—	53	2	no	—
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
105	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
107	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
108	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	M₁₆	—	106	4	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	—	106	4	no	—
114	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
115	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
116	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
117	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
118	C₄²	C₂×Dic₅₃	1	4	yes	—
119	C₄²	C₂×Dic₅₃	1	2	yes	—
120	C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄²	C₂²×D₅₃	1	2	yes	—
123	C₂³	C₂²×Dic₅₃	1	2	yes	—
124	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	#11719945	1	4	yes	—
132	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
133	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
134	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
135	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
136	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
138	C₈	—	106	4	no	—
139	C₈	—	106	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#13133176	1	2	yes	—
144	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	#13133176	1	2	yes	—
150	C₄	C₂×#11719945	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#13133176	1	1	yes	characteristic
154	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810684

C₂

yes

maximal, complemented

#16810680

C₂

yes

maximal, complemented

#16810650

C₂

yes

maximal, complemented

#16810590

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic, complemented

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₄×#41×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

#13133128

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₄

yes

—

#117×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

C₂×#41

yes

—

#2923

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#117

yes

characteristic

C₂×C₅₃

#568

yes

—

C₄×#41

D₅₃

yes

—

—

—

—

C₄×M₁₆

—

C₂×#97

—

C₅₃

#2923

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

Dic₅₃

yes

—

C₄×D₄

Dic₅₃

yes

—

C₄×D₄

Dic₅₃

yes

—

#117

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

100

#97

—

101

C₄×C₈

—

102

#41

—

103

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

104

C₂×D₄

C₂×Dic₅₃

yes

—

105

C₂×D₄

C₂×Dic₅₃

yes

characteristic

106

C₂²×C₄

C₂×Dic₅₃

yes

—

107

C₂²×C₄

C₂×Dic₅₃

yes

—

108

C₂²×C₄

C₂²×D₅₃

yes

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

M₁₆

—

106

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

C₂×Dic₅₃

yes

—

115

C₄⋊C₄

C₂×Dic₅₃

yes

—

116

C₂²⋊C₄

C₂×Dic₅₃

yes

—

117

C₂²⋊C₄

C₂×Dic₅₃

yes

—

118

C₄²

C₂×Dic₅₃

yes

—

119

C₄²

C₂×Dic₅₃

yes

—

120

C₄²

C₂²×D₅₃

yes

characteristic

121

C₄²

C₂²×D₅₃

yes

characteristic

122

C₄²

C₂²×D₅₃

yes

—

123

C₂³

C₂²×Dic₅₃

yes

—

124

C₂³

C₂²×Dic₅₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#11719945

yes

—

132

C₂×C₄

C₂²×Dic₅₃

yes

—

133

C₂×C₄

C₂²×Dic₅₃

yes

—

134

C₂×C₄

C₂²×Dic₅₃

yes

—

135

C₂×C₄

C₂²×Dic₅₃

yes

—

136

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

137

C₂×C₄

C₂³×D₅₃

yes

characteristic

138

C₈

—

106

—

139

C₈

—

106

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13133176

yes

—

144

C₂²

C₂³×Dic₅₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#13133176

yes

—

150

C₄

C₂×#11719945

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#13133176

yes

characteristic

154

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×#41×C₅₃	C₂
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#2923
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₄×#41×C₅₃

C₂

hypercenter

C₄×#41

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. #16810684	C₂	semidirect
2. #16810680	C₂	semidirect
3. #16810650	C₂	semidirect
5. C₄×#11719917	C₂	semidirect
6. C₂×#13133128	C₂	semidirect
85. C₅₃	#2923	semidirect

H ≅ G/N

type

1. #16810684

C₂

semidirect

2. #16810680

C₂

semidirect

3. #16810650

C₂

semidirect

5. C₄×#11719917

C₂

semidirect

6. C₂×#13133128

C₂

semidirect

85. C₅₃

#2923

semidirect

Subgroups — #20283366

Conjugacy classes of subgroups

Complements